Conventionally, there is a technology to encode an input signal having a plurality of channels based on spatial information. As one example of encoding an audio signal, for example, there is a parametric stereo coding technology. The parametric stereo coding technology is employed by High-Efficiency Advanced Audio Coding (HE-AAC) version 2 (hereinafter, called HE-AACv2) of Moving Picture Experts Group (MPEG)-4 audio standard (ISO/IEC 14496-3) specified by International Organization for Standardization (ISO)/International Electrotechnical Commission (IEC). The parametric stereo coding technology uses the following four types of spatial information: Inter-channel Intensity Differences (IID) that is an intensity difference between channels of an input signal, Inter-channel Coherence (ICC) that is similarity between channels of an input signal, Inter-channel Phase Differences (IPD) that is a phase difference between channels of an input signal, and Overall Phase Differences (OPD) that is a phase difference between original sound (an input signal before encoding) and a monaural signal.
Meanwhile, a technology that decodes a signal encoded by the parametric stereo coding technology is standardized by MPEG-4 audio standard (ISO/IEC 14496-3). The standardized decoding technologies include a decoding technology that uses the above-described four types of spatial information (Unrestricted version, hereinafter called a full specification version) and that uses the above-described two types of spatial information that are IID and ICC to achieve low amount of calculation (Baseline version, hereinafter called a simplified version). The decoding process of the full specification version is represented by the following expression (1). The decoding process of the simplified specification version is represented by the following expression (2).
                                              ⁢                  Expression          ⁢                                          ⁢          1                                                                              [                                                    L                                                                    R                                              ]                =                                                            [                                                                                                    c                        2                                                                                    0                                                                                                  0                                                                                      c                        1                                                                                            ]                            ⁡                              [                                                                                                    cos                        ⁡                                                  (                          α                          )                                                                                                                                    sin                        ⁡                                                  (                          α                          )                                                                                                                                                                        cos                        ⁡                                                  (                                                      -                            α                                                    )                                                                                                                                    sin                        ⁡                                                  (                                                      -                            α                                                    )                                                                                                                    ]                                      [                                                  ⁢                                                                                ⅇ                                          j                      ⁢                                                                                          ⁢                      OPD                                                                                        0                                                                              0                                                                      ⅇ                                          j                      ⁡                                              (                                                  IPD                          -                          OPD                                                )                                                                                                                  ]                    [                                          ⁢                                                    M                                                                    D                                              ]                                    (        1        )                                                          ⁢                  Expression          ⁢                                          ⁢          2                                                                                              ⁢                              [                                                            L                                                                              R                                                      ]                    =                                                    [                                                                                                    c                        2                                                                                    0                                                                                                  0                                                                                      c                        1                                                                                            ]                            ⁡                              [                                                                                                    cos                        ⁡                                                  (                          α                          )                                                                                                                                    sin                        ⁡                                                  (                          α                          )                                                                                                                                                                        cos                        ⁡                                                  (                                                      -                            α                                                    )                                                                                                                                    sin                        ⁡                                                  (                                                      -                            α                                                    )                                                                                                                    ]                                      ⁡                          [                                                                    M                                                                                        D                                                              ]                                                          (        2        )            
In the expressions (1) and (2), the L is a signal of an L channel of an audio signal, while the R is a signal of an R channel of the audio signal. The M indicates a monaural signal of the audio signal, and the D indicates a reverberation signal of the audio signal. The c1 is represented by the following expression (3). The c2 is represented by the following expression (4). The c in the expression (3) and the expression (4) is represented by the following expression (5). In the expression (5), the IID is an intensity difference between the channels. The IID is represented by the following expression (6). In the expression (6), the eL is a self correlation of the L channel signal and the eR is a self correlation of the R channel signal.
                    Expression        ⁢                                  ⁢        3                                                                      c          1                =                              2                                              1              +                              c                2                                                                        (        3        )                                Expression        ⁢                                  ⁢        4                                                                      c          2                =                                            2                        ⁢            c                                              1              +                              c                2                                                                        (        4        )                                Expression        ⁢                                  ⁢        5                                                            c        =                  10                      IID            20                                              (        5        )                                Expression        ⁢                                  ⁢        6                                                            IID        =                  10          ⁢                                          ⁢                                    log              10                        (                                          e                L                                            e                R                                      )                                              (        6        )            
The “α” in the expressions (1) and (2) is represented by the following expression (7). The “α0” in the expression (7) is represented by the following expression (8). In the expression (8), the ICC is similarity between the channels. The ICC is represented by the following expression (9). In the expression (9), the eLR is a cross correlation between the L-channel signal and the R-channel signal.
                    Expression        ⁢                                  ⁢        7                                                                      α          =                                                    α                0                            +                                                                    α                    0                                    ⁡                                      (                                                                  c                        1                                            -                                              c                        2                                                              )                                                                    2                                                      =                                          (                                  1                  +                                                            (                                                                        c                          1                                                -                                                  c                          2                                                                    )                                                              2                                                                      )                            ⁢                              α                0                                                    ⁢                                  ⁢                  Expression          ⁢                                          ⁢          8                                    (        7        )                                                      α            0                    =                                    1              2                        ⁢                          arccos              ⁡                              (                ICC                )                                                    ⁢                                  ⁢                  Expression          ⁢                                          ⁢          9                                    (        8        )                                ICC        =                                                        e              LR                                                                                      e                L                            ⁢                              e                R                                                                        (        9        )            
In the expression (1), the IPD is a phase difference between the channels. The IPD is represented by the following expression (10). The OPD is a phase difference between the original sound and the monaural signal. The OPD is represented by the following expression (11). In the expression (11), eLM is a cross correlation between the L channel signal of the original sound and the monaural signal. The monaural signal is obtained by down-mixing the L channel signal and the R channel signal of the original sound. In the expressions (10) and (11), the “Re” indicates a real part while “Inn” indicates an imaginary part.
                    Expression        ⁢                                  ⁢        10                                                                      IPD          =                                    ∠              ⁢                                                          ⁢                              e                LR                                      =                          arctan              ⁡                              (                                                      Im                    ⁡                                          (                                              e                        LR                                            )                                                                            Re                    ⁡                                          (                                              e                        LR                                            )                                                                      )                                                    ⁢                                  ⁢                  Expression          ⁢                                                            ⁢                                                          ⁢          11                                    (        10        )                                OPD        =                              ∠            ⁢                                                  ⁢                          e              LM                                =                      arctan            ⁡                          (                                                Im                  ⁡                                      (                                          e                      LM                                        )                                                                    Re                  ⁡                                      (                                          e                      LM                                        )                                                              )                                                          (        11        )            
According to the expressions (9) and (10), similarity between the channels the ICC, and a phase difference between the channels, the IPD include a cross correlation eLR between the L channel signal and the R channel signal. In other words, both the similarity between the channels (ICC), and the phase difference between the channels (IPD) include phase information. Accordingly, phase information included in the phase difference between the channels (IPD), and phase information included in the similarity between the channels (ICC) is redundantly added to signals decoded by using the full specification decoding technology. As a result, signals decoded by the full specification version differ from the signals before encoding. Thus, there is a method to generate similarity between the channels (ICC) without including the phase information. When similarity between the channels (ICC) does not include the phase information, signals before encoding may be reproduced by the full specification version decoding technology.